reserve h,r,r1,r2,x0,x1,x2,x3,x4,x5,x,a,b,c,k for Real,
  f,f1,f2 for Function of REAL,REAL;

theorem
  for x holds cD(sin(#)cos,h).x = (1/2)*(sin(2*x+h)-sin(2*x-h))
proof
  let x;
  cD(sin(#)cos,h).x = (sin(#)cos).(x+h/2) - (sin(#)cos).(x-h/2) by DIFF_1:5
    .= (sin.(x+h/2))*(cos.(x+h/2)) -(sin(#)cos).(x-h/2) by VALUED_1:5
    .= sin(x+h/2)*cos(x+h/2) -sin(x-h/2)*cos(x-h/2) by VALUED_1:5
    .= (1/2)*(sin((x+h/2)+(x+h/2))+sin((x+h/2)-(x+h/2))) -sin(x-h/2)*cos(x-h
  /2) by SIN_COS4:30
    .= (1/2)*(sin((x+h/2)+(x+h/2))+sin((x+h/2)-(x+h/2))) -(1/2)*(sin((x-h/2)
  +(x-h/2)) +sin((x-h/2)-(x-h/2))) by SIN_COS4:30
    .= (1/2)*sin(2*(x+h/2))-(1/2)*sin(2*(x-h/2));
  hence thesis;
end;
